Finite Group Databasebeta

← #164172

Subgroups — #164172

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36259C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic, complemented
3#20336C211yesmaximal, characteristic, complemented
4#7685C211yesmaximal, characteristic, complemented
5#7678C212yesmaximal, complemented
6#7668C211yesmaximal, characteristic, complemented
7#7654C212yesmaximal, complemented
8#7616C211yesmaximal, characteristic, complemented
9#7610C212yesmaximal, complemented
10#7565C211yesmaximal, characteristic, complemented
11C2×#1271C212yesmaximal, complemented
12C22×#131—21no—
13C2×#547—21no—
14C2×#547C411yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C2×C8×D4—21no—
17#2857C2211yescharacteristic
18#2854C2212yes—
19#2546C411yescharacteristic
20#2538C411yescharacteristic
21#2486C412yes—
22#2429—21no—
23#2425—22no—
24#2416C412yes—
25#2415—21no—
26#2413—22no—
27D4×C4⋊C4C411yescharacteristic
28#2339—21no—
29C4×#121C2211yescharacteristic
30C2×#389C2212yes—
31#1984—21no—
32#1869—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35C4×C22⋊C8C2211yescharacteristic
36#1729—22no—
37#1729—21no—
38#1728C2211yescharacteristic
39#1728C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41—21no—
54C22×#41D411yescharacteristic
55C2×#131—28no—
56C2×#131—44no—
57C2×#131—44no—
58C22×M16—22no—
59C22×M16—21no—
60C23×C8—22no—
61C23×C8—21no—
62C23×C8—22no—
63#547—44no—
64#547—44no—
65#547—44no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#123—22no—
69C2×#123—22no—
70C2×#123—22no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×#121—22no—
75C2×#121C2×C411yescharacteristic
76C2×#121C2×C412yes—
77C4×#41—44no—
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2×C412yes—
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D412yes—
87C2×#117D412yes—
88C2×#117D411yescharacteristic
89C22×C4⋊C4—22no—
90C22×C4⋊C4C2×C412yes—
91C22×C4⋊C4C2×C412yes—
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C8×D4—44no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C4×C8—21no—
100#398—22no—
101#398—22no—
102#397—22no—
103#396—22no—
104#396—22no—
105#394—22no—
106#391—21no—
107#391—21no—
108#390C2×C412yes—
109#390C2311yescharacteristic
110#389—21no—
111#389—21no—
112#389—22no—
113#389—22no—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2×C412yes—
117#389C2312yes—
118#389C2311yescharacteristic
119#388—22no—
120#388—22no—
121#388—21no—
122#386—22no—
123#386—21no—
124#386—22no—
125#385—22no—
126#384C2×C411yescharacteristic
127#381—22no—
128#380—21no—
129#380—21no—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C22⋊C4—21no—
135C2×#95—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—21no—
140#336—21no—
141#336—22no—
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146#336C2312yes—
147#336C2311yescharacteristic
148C2×#41—24no—
149C2×#41—24no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C2×#41D814yes—
154C2×#41SD1614yes—
155C22×Q8—21no—
156C22×Q8C22⋊C411yescharacteristic
157C22×D4—22no—
158C22×D4—21no—
159C22×D4C22⋊C412yes—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171#131—432no—
172C2×M16—28no—
173C2×M16—24no—
174C2×M16—44no—
175C2×M16—42no—
176C2×M16—44no—
177C2×M16—42no—
178C22×C8—28no—
179C22×C8—24no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—44no—
187C22×C8—44no—
188C22×C8—42no—
189C22×C8—44no—
190C22×C8—42no—
191C22×C8—44no—
192C22×C8—82no—
193C4⋊Q8—42no—
194C4⋊Q8—44no—
195#123—48no—
196#123—48no—
197#123—44no—
198#123—44no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203#121—22no—
204#121—41no—
205#121—44no—
206#121—48no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—42no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—42no—
216C4×D4—42no—
217#117—44no—
218#117—42no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C22⋊C4C22⋊C412yes—
256C2×C42—21no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2411yescharacteristic
266C4⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—44no—
269C22⋊C8—81no—
270C4×C8—42no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—21no—
279#95—22no—
280C24—21no—
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C412yes—
283#41—216no—
284#41—216no—
285#41—816no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8#10214yes—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—81no—
304C2×D4#9914yes—
305C2×D4#10218yes—
306C2×D4#10214yes—
307C2×D4#10414yes—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—82no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×D814yes—
349C22×C4C2×D812yes—
350C22×C4C2×SD1614yes—
351C22×C4C2×SD1612yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355M16—416no—
356M16—48no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—416no—
366C2×C8—48no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—88no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—81no—
400C4⋊C4#13112yescomplemented
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—44no—
415C22⋊C4—82no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—42no—
420C42—41no—
421C42—42no—
422C42—42no—
423C42—42no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23C2×#9914yes—
442C23C2×#10212yes—
443C23C2×#10214yes—
444C23C2×#10414yes—
445C23C22×C22⋊C411yescharacteristic
446Q8—28no—
447Q8—84no—
448D4—216no—
449D4—81no—
450D4—81no—
451D4—84no—
452D4—88no—
453D4#327116yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—88no—
502C2×C4—88no—
503C2×C4—84no—
504C2×C4C2×#9912yes—
505C2×C4C2×#10214yes—
506C2×C4C2×#10212yes—
507C2×C4C2×#10412yes—
508C2×C4C2×#13111yescharacteristic
509C2×C4#56811yescharacteristic
510C2×C4C22×D811yescharacteristic
511C2×C4C22×SD1611yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57811yescharacteristic
514C8—42no—
515C8—44no—
516C8—42no—
517C8—42no—
518C8—81no—
519C8—82no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C2×#32718yes—
541C22#288011yescharacteristic
542C22C22×#9911yescharacteristic
543C22#288911yescharacteristic
544C22C22×#10211yescharacteristic
545C22#289811yescharacteristic
546C22C22×#10411yescharacteristic
547C22#290111yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—24no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—81no—
561C4—82no—
562C4—81no—
563C4—82no—
564C4C2×#32714yes—
565C2—28no—
566C2—42no—
567C2—161no—
568C2#1741611yescharacteristic
569C2#1750711yescharacteristic
570C2#1751911yescharacteristic
571C2C22×#32711yescharacteristic
572C2#1760811yescharacteristic
573C2#1765311yescharacteristic
574C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36259C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22469C2semidirect
3. #20336C2semidirect
4. #7685C2semidirect
5. #7678C2semidirect
6. #7668C2semidirect
7. #7654C2semidirect
8. #7616C2semidirect
9. #7610C2semidirect
10. #7565C2semidirect
11. C2×#1271C2semidirect
400. C4⋊C4#131semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#362592
#224692
#203362
#76852
#76782
#76682
#76542
#76162
#76102
#75652
C2×#12712